Let $a$ and $b$ be positive real numbers such that $a + b = 1.$  Find set of all possible values of $\frac{1}{a} + \frac{1}{b}.$
Answer: By AM-HM,
\[\frac{a + b}{2} \ge \frac{2}{\frac{1}{a} + \frac{1}{b}}.\]Hence,
\[\frac{1}{a} + \frac{1}{b} \ge \frac{4}{a + b} = 4.\]Equality occurs when $a = b = \frac{1}{2}.$

Note that as $a$ approaches 0 and $b$ approaches 1, $\frac{1}{a} + \frac{1}{b}$ becomes arbitrarily large.  Therefore, the set of all possible values of $\frac{1}{a} + \frac{1}{b}$ is $\boxed{[4,\infty)}.$